Elastic modulus plays a central role in the understanding of the mechanical behavior of a material. In practice, there is a need to determine the elastic modulus of materials on small scales. In the past decade, depth-sensing indentation technique has become a very useful tool for this purpose, particularly in determining the mechanical properties of materials on small scales by recording the load v.s. displacement of the indenter during indentation (loading and unloading curves), from which the mechanical properties of the indented material are estimated.
Oliver and Pharr (J. Mater. Res. 7, 1564, (1992)) proposed a classic formula correlating the reduced elastic modulus (Er), the initial slope of the unloading curve Su, and the projected contact area A(hc) at the maximum indentation load:
                              E          r                =                              π                          2              ⁢                                                          ⁢              β                                ⁢                      (                                          S                u                                                              A                  ⁡                                      (                                          h                      c                                        )                                                                        )                                              (        1        )            A(hc) is the cross sectional area of the indenter corresponding to the contact depth hc at the maximum indentation load, as shown in FIG. 1. Er is defined by the expression
      1          E      r        =                    1        -                  v          2                    E        +                  1        -                  v          i          2                            E        i            with E and v being the elastic modulus and Poisson's ratio of indented material, and Ei and vi being those of the indenter. β is an indenter shape dependent constant. In this method, A(hc) is estimated indirectly from the unloading curve in order to avoid direct imaging of the impression. As such, errors could be introduced, especially when “piling-up” of the indented material at the point of contact occurs. This situation of “piling-up” is shown in FIG. 2. Moreover, the initial unloading slope Su of the unloading curve is needed, but it is sometimes difficult to be determined accurately, especially in the cases where the signal-to-noise ratio is low. Because this method requires the use of Su, it is therefore referred as the slope method in the context.
Regarding the above deficiencies, in recent years, some researchers sought for alternative approaches, such as examining the relationship between hardness, elastic modulus and indentation work on the basis of numerical simulation for ideally sharp indentation. It was found (Y.-T. Cheng and C.-M. Cheng, Appl. Phys. Lett. 73, 614(1998)) that the ratio of hardness (H) to reduced elastic modulus can be related to the ratio of elastic work (We) to total work (W) in an indentation, in implicit form:H/Er=f(We/W)  (2)where H=Pm/A(hc) is measured at the maximum indentation load Pm; A(hc) is the projected contact area corresponding to the contact depth hc, as shown in FIG. 1; We and W are the work done by the indenter in the unloading and the loading processes, respectively, as shown in FIG. 3. By combining Eq(1) and Eq(2), Er can be determined as:Er=[π/(2β)2]f(We/W)[Su2/Pm]  (3)
Compared with the slope method, this method does not require A(hc), but it still relies on the use of the initial unloading slope, which may be the major source of error. Associated with these particular features, this approach is denoted as the slope&energy method in the context. Further, in Cheng et. al. paper, the indenter is assumed to be ideally sharp, so that the treatment is not detailed enough for the model to be practically useful, but more work has to be done to take the indenter bluntness effects into consideration.